Extrapolation of an analytical function

Question

Given a function $f:z\mapsto f(z)$ for a discrete set of points in the real interval $z\in[a,b]$ and the knowledge that $f$ is analytical along the real axis and that its Fourier transform is real valued along $\mathbb R$ (and thus $f(-z)=f^*(z)$), is it possible to reconstruct $f(z)$ for all $z\in\mathbb C$? If not, what approximations are possible?

Answer 1

No, it's not possible to reconstruct $f(z)$. You can augment your set of given function values by the corresponding points $-z$ to fulfill the symmetry condition, then pick an arbitrary sufficiently decaying even analytic function $g$, e.g. any Gaussian centered at $z=0$, and divide all the function values by the value of $g$ at that point. Then you can choose arbitrary polynomials that interpolate between the resulting values, and each such polynomial multiplied by $g$ will have the required properties. You have a lot of freedom of choice in $g$ and in the polynomials, so $f$ is very much not determined by these conditions.

Concerning the question what approximations are possible: For the polynomials, there's a natural choice, the unique polynomial of lowest degree that interpolates the function values divided by $g$. But there's no natural choice for $g$; different functions $g$ will lead to different interpolations for $f$, all with the required properties.